
The generation of primordial blackholes and the induced second-order gravitational waves in the natural inflation model (REF ) (without the Gauss-Bonnet correction) has been discussed, e.g., in Refs. [1]}, [2]}, [3]}, [4]}, [5]}, where the axion (inflaton) field is coupled to gauge fields.
Refs. [6]}, [7]} discuss primordial blackholes and the gravitational waves in the natural inflation model with the modified inflaton kinetic sector.
